exercise 5.8
write an informal proof of
Premises: LeftOf(a,b) | RightOf(a,b)
BackOf(a,b) | ~Leftof(a,b)
FrontOf(b,a) | ~RightOf(a,b)
SameCol(c,a) & SameRow(c,b)
conclusion BackOf(a,b)
State if you use proof by cases.
The first premiss tells us that a is either to the left of b or to the right of b. Let’s
consider each of these possibilities in turn.
1. Assume that a is to the left of b. Then, from the second premise, a must be back
of b, which is the conclusion.
2. Assume that a is to the right of b. Then, from the third premise, b must be front
of a. This is equivalent to saying that a is back of b, which is the conclusion.
Either way, then, the conclusion follows from the premises.
To write an informal proof of the conclusion "BackOf(a,b)" based on the given premises, we can follow a step-by-step approach. Here's how we can proceed:
1. Assume the premises:
- LeftOf(a,b) | RightOf(a,b)
- BackOf(a,b) | ~LeftOf(a,b)
- FrontOf(b,a) | ~RightOf(a,b)
- SameCol(c,a) & SameRow(c,b)
2. We need to show that "BackOf(a,b)" holds. To do this, we will consider two cases:
a) Assume "LeftOf(a,b)" is true.
b) Assume "RightOf(a,b)" is true.
3. Case 1: Assume "LeftOf(a,b)" is true.
- From premise 1 (LeftOf(a,b) | RightOf(a,b)), we can conclude that "LeftOf(a,b)" is true.
- From premise 2 (BackOf(a,b) | ~LeftOf(a,b)), we can infer that "BackOf(a,b)" is true because we assumed "LeftOf(a,b)".
- Therefore, in this case, we have shown that "BackOf(a,b)" holds.
4. Case 2: Assume "RightOf(a,b)" is true.
- From premise 1 (LeftOf(a,b) | RightOf(a,b)), we can conclude that "RightOf(a,b)" is true.
- From premise 3 (FrontOf(b,a) | ~RightOf(a,b)), we can infer that "~RightOf(a,b)" is true because we assumed "RightOf(a,b)".
- This leads to a contradiction because we have assumed "~RightOf(a,b)" and deduced that "RightOf(a,b)" is true.
- Therefore, this case is not possible.
5. Since we derived "BackOf(a,b)" in Case 1 (by assuming "LeftOf(a,b)"), we can conclude that "BackOf(a,b)" holds.
Therefore, we have proven "BackOf(a,b)" based on the given premises. In this proof, we used proof by cases by considering two possible options: "LeftOf(a,b)" and "RightOf(a,b)".
To write an informal proof for the given premises and conclusion, we will consider each premise and use logical reasoning to reach the desired conclusion. Let's break it down step by step:
1. Premise: LeftOf(a,b) | RightOf(a,b)
This premise states that either "a is left of b" or "a is right of b."
2. Premise: BackOf(a,b) | ~LeftOf(a,b)
This premise states that either "a is back of b" or "a is not left of b."
3. Premise: FrontOf(b,a) | ~RightOf(a,b)
This premise states that either "b is front of a" or "a is not right of b."
4. Premise: SameCol(c,a) & SameRow(c,b)
This premise states that "c is in the same column as a" and "c is in the same row as b."
Now, let's analyze the premises and derive the conclusion:
We know that if "LeftOf(a,b)" is true, then "BackOf(a,b)" must be true, based on premise 2.
Similarly, if "RightOf(a,b)" is true, then "BackOf(a,b)" must also be true, based on premise 1.
Now, we consider the case where "LeftOf(a,b)" is false. In this case, we can use premise 4 which states that "c is in the same column as a" and "c is in the same row as b." Since "c" is in the same row as "b," it can be considered "front of b." And, based on premise 3, if "FrontOf(b,a)" is true, then "~RightOf(a,b)" is also true. And since "~RightOf(a,b)" is true, "BackOf(a,b)" must be true.
Therefore, in both cases, "BackOf(a,b)" is true.
In conclusion, we have proven that "BackOf(a,b)" is true based on the given premises using cases.
We want to show that if we make the premises of the argument true, the conclusion must be true. In holding
premise